Multimodal Locomotion and Active Targeted Thermal Control of Magnetic Agents for Biomedical Applications

Abstract Magnetic microrobots can be miniaturized to a nanometric scale owing to their wireless actuation, thereby rendering them ideal for numerous biomedical applications. As a result, nowadays, there exist several mechano‐electromagnetic systems for their actuation. However, magnetic actuation is not sufficient for implementation in biomedical applications, and further functionalities such as imaging and heating are required. This study proposes a multimodal electromagnetic system comprised of three pairs of Helmholtz coils, a pair of Maxwell coils, and a high‐frequency solenoid to realize multimodal locomotion and heating control of magnetic microrobots. The system produces different configurations of magnetic fields that can generate magnetic forces and torques for the multimodal locomotion of magnetic microrobots, as well as generate magnetic traps that can control the locomotion of magnetic swarms. Furthermore, these magnetic fields are employed to control the magnetization of magnetic nanoparticles, affecting their magnetic relaxation mechanisms and diminishing their thermal properties. Thus, the system enables the control of the temperature increase of soft‐magnetic materials and selective heating of magnetic microrobots at different positions, while suppressing the heating properties of magnetic nanoparticles located at undesired areas.


Section 2. Detailed configuration of MECS
For the design of our system, we used a commercial induction heating coil (Hc) (Osung Hitech, OSH-R5), comprising a 5 kW power supply and a solenoid coil with the number of turns N = 4 and a diameter D = 6.5 cm. The Hc coil control system had a fixed frequency of 200 kHz, and only the ON/OFF and power (in percentage) could be manually controlled. To obtain the amplitude of the magnetic field, we used a pick-up coil with N = 7 and D = 2 cm to measure it indirectly using Faraday's law of induction, as shown in Figure S2. where P is the power supplied from the power supply in percentage. Considering the volume that can fit within the coil Hc, we designed the remaining coils for a working space of 60 mm x 60 mm x 60 mm. First, the pair of Maxwell coils are designed because they are in charge of creating the field-free region (FFR). They require a high current for generating a gradient above 2 mT mm −1 , which is the minimum gradient typically reported for magnetic particle imaging applications and available in commercial magnetic particle imaging systems (24). We designed our pair of Maxwell coils for a magnetic gradient field of G = [-1.5 -1. 5 3] mT mm −1 .   Figure S3 and Figure S4.  The magnetic field produced by the three pairs of Helmholtz coils is: where n, I, and R are the number of turns, electric current, and radius of each coil, respectively.
Because n and I are constants, we can simplify the expression as: where kx, ky, and kz are constants that depend on the properties of the coils. The magnetic field created by the pair of Maxwell coils is: where G is the 3 x 3 diagonal gradient matrix: where km is a constant that is dependent on the coil geometrical properties. All the constants kx, ky, kz, and km can be calculated; however, for consistency between our simulations and the real magnetic field distribution, we obtained them experimentally.
Using a gaussmeter (5180, F.W. Bell, USA), we measured the magnetic field produced by the Helmholtz coils at its center for different values of electric current. Similarly, we measured the magnetic field produced by the pair of Maxwell coils along the Z and X axis and calculated its magnetic gradient, as shown in Figure S5. The gradient produced in the X axis is the same as that in the Y axis and the gradient produced along any radial axis perpendicular to the Z axis will be same. Thus, we can describe the gradient distribution only through two terms Gz and Gr . We compared the measured data with the simulated values and slightly modified the simulation parameters such that the simulated values matched the measured ones, to enable higher precision when calculating the FFR. Through these measurements and simulations, we determined the following values: kx = 1.76 x 10 -3 , ky = 1.8 x 10 -3 , kz = 3.74 x 10 -3 , and km = 0.035.  Maxwell coils for the control of magnetic microrobots. When comparing MECS even with the recently reported systems (2020-2021), it can be observed that MECS has the biggest working space, produces the highest uniform magnetic fields and magnetic field gradients, and implements close-loop multimodal locomotion of single magnetic microrobots; it is the only system that can produce magnetic traps for the self-assembly and control of magnetic swarms.
Moreover, it performs temperature control of MNP and selective heating of MNP.
8  was used to test the locomotion of helicoidal microrobots through the application of uniform rotating magnetic fields. For magnetic force locomotion, we used robot MR2 ( Figure S6B). As shown in Figure S6C, for M3 we placed a neodymium magnet inside an acrylic disk and used it for locomotion based on the trapping point mechanism. Further, for the locomotion and eddycurrent heating experiments, we used MR4 ( Figure S6D). MR5 was used for the locomotion and heating control by FFR experiments. In addition, the magnetic swarm MS1 ( Figure S6F) was composed of seven identical MR3 robots. MR6 ( Figure S6G) was used for the exemplification of targeted therapy and locomotion in an artificial body fluid (plasma fluid). Figure S7. Forces acting on a helicoidal magnetic microrobot.

Section 4. Modeling for magnetic locomotion
The magnetic torque exerted in a magnetic microrobot is: where, The RMF is controlled using the following equation: where θ is the azimuthal angle, and Ψ is the polar angle of the axis of the RMF. Assuming that the RMF has a sufficiently large magnitude such that the step-out frequency of the microrobot for a particular fluid is beyond the working range, the thrust force FT in a helicoidal magnetic microrobot is proportional to the angular frequency ω; thus, we obtain where λ is a constant that depends on the geometric properties of the robot and the fluid.
Considering that the robot experiences a gravity force, Fg, that causes it to move to the bottom of the container, to make the robot maintain its position, we must insert a compensation thrust force FTg. Consequently, for the robot to swim in any desired direction φ (of the zr plane) an additional locomotion thrusting force FL must be produced.
λ and ω0 can be calculated using a set of complex equations, but the results may vary significantly from the real results. Hence, it is better to determine λ and ω0 experimentally, which can be easily done using Equation S1. Thereafter, the user can control the locomotion of the microrobot by simply indicating the desired locomotion direction (φ, θ), force, and speed or locomotion frequency (f). Subsequently, the control algorithm calculates the required total angular frequency of the magnetic field as well as its rotation axis. Figure S8. Magnetic force locomotion. A) Locomotion mechanism using magnetic force. B) Forces acting on the magnetic microrobot.
For the magnetic force locomotion, we used the pair of coils Hz to create a three-dimensional 1 DOF magnetic gradient field, as shown in Figure S8A, which drags the microrobot along the mirrored axis of the magnetic moment of the robot with respect to the axis of the coils Hz. Figure S8B shows the forces that act on the microrobot as it moves through a fluid. The microrobot experiences a force Fg due to gravity, a buoyancy force Fb that causes the magnet to move to the surface of the liquid, and a drag force Fd that opposes the movement of the microrobot in the fluid. We define Fc and Fl as the gravity compensation and magnetic locomotive forces, respectively, produced by the coil system. The magnetic force exerted on a microrobot by the pair of gradient coils is Because of the geometry of the gradient produced by the Hz coils, we analyze the magnetic 12 force in the ZR plane, where z is the vertical axis and r is any axis perpendicular to z. Redefining , its contribution to the overall magnetic force can be neglected, resulting in: At equilibrium, the vertical forces are balanced, and . This equation can be solved to determine the value of gz required to keep the microrobot floating, yielding To drive the microrobot, we include an additional magnetic force for the propulsion of the (S20) Consequently, we can control the motion and the orientation of a microrobot in the threedimensional space (x, y, z) using magnetic forces, by controlling the magnitude of the magnetic field gradient produced by the Hz coils, and the direction of the magnetic field, with the following expression   cos cos cos sin sin Locomotion experiments herein aimed to demonstrate the different types of automatic locomotion that can be implemented using MECS. Therefore, the above equations were used for a simple implementation of closed-loop controllers. However, for the development of a more precise automatic locomotion control algorithm, additional forces such as wall effects and interactions between the robots must be considered. For instance, the drag force for a spherical particle in a flowing fluid, including wall effects, is given by [6]  where ρf is the fluid density, A is the frontal area of the robot, Cd is the drag coefficient, v is the velocity of the robot, vf is the fluid's velocity, λ= 2r/D is the robot diameter ratio with D being the vessel (or channel) diameter, and λ0 and α0 are the functions of Reynolds number, commonly set to 1.5 and 0.29, respectively.
The magnetic dipolar interactions between two robots α and β can be calculated as [7]          where m is the respective magnetic moment, and rαβ is the distance between the robots.
14 Section 5. Detailed temperature measurements of temperature increase control of magnetic nanoparticles experiments Figure S9. Temperature increase of MF1. Temperature increase for the five samples of MF1 when exposed to different values of SMF.
For analyzing the temperature increase (∆T) of MF1 when exposed to an AMF of 15.6 kA m −1 and SMF values ranging from 0 to 32 kA m −1 , we prepared five vials (S1 to S5) with 2 ml of MF1 each. As can be observed in Figure S9, the ∆T was similar for S2, S3 and S4, but it varied significantly for S1 and S5. Although we used a tip sonication to disperse the fluid before the experiments, it appears that the MNP were not evenly distributed along the MF1, which explains the variability in the data. However, when we normalized ∆T according to the maximum ∆T of each sample, we observed similar trends. Thus, we normalized the ∆T values and considered the average to create the graph shown in Figure 5D. Figure S10 shows the temperature measured in ºC at each point in the acrylic container that was filled with MF2, from P1 to P13. As can be observed, the temperature shows a similar decrease in the measured temperature with the increase in the value of the applied SMF. The temperature of the points found at the edge was higher because the magnetic field produced by the induction coil was slightly higher at that location. In addition, the temperature at P4 is significantly lower than the temperature measured at the other points, suggesting that the concentration at that point was lower. However, P4 followed the same reduction in the temperature with the increase in the SMF value. Figure S10. Temperature increase of MF2. Temperature measured at each point containing MF2 when exposed to an SMF of 0, 4, and 8 kA m −1 . Figure S11 and S12 show the temperature measured at each point containing MF2 for the experiments on focused heating using an FFR. Figure S11 shows the temperature when controlling the size of the FFR through the value of the magnetic gradient ∇Br (Gr), whereas Figure S12 shows the temperature when the position of the FFR was changed. Figure S11. FFR size control. Temperature measured at each point containing MF2 when exposed to an ∇Br of 0, 0.32, and 0.65 mT mm −1 . Figure S12. FFR position control. Temperature measured at each point containing MF2 when the FFR is moved from P7 to P2 and P9.

Supporting Videos
Video S1: Magnetic torque and magnetic force locomotion control.
Video S2: Single and collective locomotion of microrobots using trapping point.
Video S3: Temperature control of MF1 using SMF.
Video S4: FFR control-based focused heating of MNP.
Video S5: Selective melting of magnetic jellies using SMF.
Video S6: Hard-magnetic microrobot locomotion and eddy currents heating with suppression of MNP heating using an SMF.
Video S7: Soft-magnetic microrobot locomotion and targeted heating using FFR.
Video S8: Exemplification of targeted therapy in an artificial body fluid.